The sharp comparison conjecture for the triangular ratio and $w_{\mathbb{B}^2}$ metrics

Let $\mathbb{B}^2$ be the unit disk, and let $s_{\mathbb{B}^2}$ and $w_{\mathbb{B}^2}$ denote the triangular ratio metric and the function defined in the paper, respectively. The sharp comparison conjecture for the triangular ratio and $w_{\mathbb{B}^2}$ metrics. For all $x,y\in\mathbb{B}^2$, the inequality

$$s_{\mathbb{B}^2}(x,y)\leq c\,w_{\mathbb{B}^2}(x,y)$$

holds with the sharp constant

$$c=\sqrt{\frac{h_0^2-2h_0+2}{2h_0^2-2\sqrt{2}h_0+2}}\approx1.07313,\qquad h_0=\frac{1-\sqrt{9-6\sqrt{2}}}{2-\sqrt{2}}.$$

The inequality is supported by numerical tests beyond the special case proved earlier, and the stated sharp constant and its universal validity remain open.